Solve each of the following system of equations in R . 2 x - 3 4 - 2 ≥ 4 x 3 - 6 , 2 ( 2 x + 3 ) < 6 x - 2 + 10

First inequation is 2 x - 3 4 - 2 ≥ 4 x 3 - 6 ⇒ 2 x - 3 - 8 4 ≥ 4 x - 18 3 ⇒ 6 x - 22 ≥ 16 x - 54 ⇒ 10 x ≤ 32 ⇒ x ≤ 16 5 ⇒ x ∈ - ∞ , 16 5 Second inequation is 2 2 x + 3 < 6 x - 2 + 10 ⇒ 4 x + 6 < 6 x - 12 + 10 ⇒ 6 + 2 < 6 x - 4 x ⇒ 2 x > 8 ⇒ x > 4 ⇒ x ∈ 4 , ∞ ∴ Solution set of given system of inequations is the intersection of solution of 1 s t and 2 n d inequations ∴ x ∈ - ∞ , 16 5 ∩ 4 , ∞ = ∅ Hence, given system of inequations has no solution.

Solve each of the following system of equations in R . 7 x - 1 2 < - 3 , 3 x + 8 5 + 11 < 0

First equation is 7 x - 1 2 < - 3 ⇒ 7 x - 1 < - 6 ⇒ 7 x < - 5 ⇒ x < - 5 7 ⇒ x ∈ - ∞ , - 5 7 Second equation is 3 x + 8 5 + 11 < 0 ⇒ 3 x + 8 + 55 < 0 ⇒ 3 x < - 63 ⇒ x < - 21 ∴ x ∈ - ∞ , - 21 ∴ Solution set of given system of inequations is the intersection of solution of 1 s t and 2 n d inequation ∴ x ∈ - ∞ , - 5 7 ∩ - ∞ , - 21 ⇒ x ∈ ( - ∞ , - 21 )

Solve each of the following system of equations in R : 2 x + 1 7 x - 1 > 5 , x + 7 x - 8 > 2

First equation is 2 x + 1 7 x - 5 - 5 > 0 ⇒ 2 x + 1 - 35 x + 25 7 x - 5 > 0 ⇒ - 33 x + 26 7 x - 5 > 0 ⇒ 33 x - 26 7 x - 5 < 0 Case i ) 33 x - 26 < 0 and 7 x - 5 > 0 ⇒ 33 x < 26 and 7 x > 5 ⇒ x < 26 33 and x > 5 7 ⇒ 5 7 < x < 26 33 Case i i ) 33 x - 26 > 0 and 7 x - 5 < 0 ⇒ x > 26 33 and x < 5 7 This case has no solution. ∴ x ∈ 5 7 , 26 33 Second equation is x + 7 x - 8 > 2 ⇒ x + 7 x - 8 - 2 > 0 ⇒ x + 7 - 2 x + 16 x - 8 > 0 ⇒ - x + 23 x - 8 > 0 ⇒ x - 23 x - 8 < 0 Case i ) x - 23 < 0 and x - 8 > 0 ⇒ x < 23 and x > 8 ⇒ 8 < x < 23 Case i i ) x - 23 > 0 and x - 8 < 0 ⇒ x > 23 and x < 8 This case has no solution. ∴ x ∈ 8 , 23 Solution set of given system of inequations is the intersection of solution of 1 s t and 2 n d inequation ∴ x ∈ 5 7 , 26 33 ∩ 8 , 23 = ϕ Hence, the given system of inequations has no solution.